7 December 2023, 2pm
Geometry & AI
In this talk, we will present some applications of finite-dimensional and infinite-dimensional geometry in Artificial Intelligence. The growing need to process geometric data such as curves, surfaces or fibered structures in a resolution-independent way that is invariant to shape-preserving transformations gives rise to mathematical questions both in pure and applied differential geometry, but also in numerical analysis. To illustrate this, applications of shape analysis in medical imaging and temporal alignement of Human motions will be given. On the other hand, the extraction of geometric information from point clouds formed by data sets is an area where differential geometry meets probability, as in dimension reduction or manifold learning. Some of the challenges in these areas will be mentioned.
The wald space for phylogenetic trees
Most existing metrics between phylogenetic trees directly measure differences in topology and edge weights, and are unrelated to the models of evolution used to infer trees. We describe metrics which instead are based on distances between the probability models of discrete or continuous characters induced by trees. We describe how construction of information-based geodesics leads to the recently proposed wald space of phylogenetic trees. As a point set, it sits between the BHV space (Billera, Holmes and Vogtmann, 2001) and the edge-product space (Moulton and Steel 2004). It has a natural embedding into the space of positive definite matrices, equipped with the information geometry. Thus, singularities such as overlapping leaves are infinitely far away, proper forests, however, comprising the “BHV-boundary at infinity”, are part of the wald space, adding boundary correspondences to groves (corresponding to orthants in the BHV space). In fact the wald space contracts to the complete disconnected forest. Further, it is a geodesic space, exhibiting the structure of a Whitney stratified space of type (A) where strata carry compatible Riemannian metrics. We explore some more geometric properties, but the full picture remains open. We conclude by identifying open problems, we deem interesting. This is joint work with Tom Nye, Jonas Lueg, Maryam Garba.
- Billera, L., S. Holmes, and K. Vogtmann (2001): Geometry of the space of phylogenetic trees. Advances in Applied Mathematics 27 (4), 733–767.
- Garba, M. K., T. M. Nye, J. Lueg, and S. F. Huckemann (2021): Information geometry for phylogenetic trees. Journal of Mathematical Biology 82(3), 1–39.
- Lueg, J., M. K. Garba, T. M. Nye, J. Lueg, and S. F. Huckemann (2022): Foundations of wald space for statistics of phylogenetic trees. arXiv 2209.05332.
- Moulton, V. and M. Steel (2004): Peeling phylogenetic ’oranges’. Advances in Applied Mathematics 33(4), 710–727.
14 December 2023, 2pm
Software tools to learn shape manifolds
3D shapes are best understood as samples on a curved manifold. Can we design robust and interpretable methods that “unfold” high-dimensional spaces of anatomical shapes with a minimal amount of training and parameter tuning? Going beyond tangent approximations of standard metrics, I will present fast numerical tools that enable non-linear manifold learning with real data. Notably, I will showcase a beta version of the “scikit-shapes” Python library and discuss the interaction of standard dimensionality reduction algorithms with the Wasserstein metric that is induced by optimal transport.